Multiply: (x + 6)(3x² ­- 2x + 5)?

Multiplying polynomials is relatively "easy", even if it can be long and tedious. The process is called a "distribution" of factors.

Each term in one polynomial multiplies each term in the other one.

Here is the step-by-step, slomo replay:

(x + 6)(3x^2 - 2x + 5)

You can pick any one as the one you "break down" for distribution; I normally pick the shorter one.

Each term of (x+6) will multiply the other polynomial:

(x)(3x^2 - 2x + 5) + (6)(3x^2 - 2x + 5)

now, distribute each factor into the bracket - keep track of signs:
(x)(3x^2) + (x)(-2x) + (x)(5) + (6)(3x^2) + (6)(-2x) + (6)(5)

perform the individual multiplications - keep track of signs:

3x^3 - 2x^2 + 5x + 18x^2 - 12x + 30

clean up, regroup similar powers:
3x^3 + 18x^2 - 2x^2 + 5x - 12x + 30
3x^3 + 16x^2 - 7x + 30

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When using a GCF (also called a GCD), you can always do a "prime factorization", whether you are working with numbers, terms, or whatever other objects.

Here, the prime factorization of each term looks like this (I always begin with numbers, then variables)

12x^4 = 2*2*3 * x*x*x*x
16x^3 = 2*2*2*2 * x*x*x
48x^2 = 2*2*2*2*3 * x*x

For the GCF, you can only take things that are common to ALL of them.
Here, we can pick two "2"s and two "x"s, giving us
2*2 * x*x = 4x^2
(we cannot pick the "3" because it is missing in one of them)
(we cannot pick "a third x" because that 3rd x is missing in the last term)

4x^2
That is our GCF
and that is the "common factor" that you can factor out:

(12x^4 + 16x^3 - 48x^2)
=(4x^2)( divide each term by 4x^2 )

If there are TWO questions , number them as 1 and 2.

1
3x³ - 2x² + 5x + 18x² - 12x + 30
3x³ + 16x² - 7x + 30

2.
GCF is 4x^2

(x+6)(3x^2-2x+5)
(x)(3x^2) + (x)(-2x) + (x)(5) + (6)(3x^2)+(6)(-2x)+(6)(5)
3x^3-2x^2+5x+18x^2-12x+30
3x^3 +16x^2-7x+30

12x^4+16x^3-48x^2
4x^2 (3x^2+4x-12)